Akhiezer damping and the thermal conductivity of pure and impure dielectrics

The ultrasonic attenuation of pure and impure dielectric crystals in the high-temperature or Akhiezer regime is derived in a relaxation-time approximation which includes both dispersion and anisotropy of all relevant thermal phonon parameters. The attenuation coefficient results are proportional to the product of the thermal conductivity for temperature gradients along the sound-wave propagation direction Kₖ times an average of the phonon-mode Gr\"uneisen parameter weighted by Kₖ. Experimental data of the attenuation in the ultrahigh-frequency band at temperatures between 80 and 300 K, in pure crystals of Si, Ge, TiO₂, MgO, and SiO₂, are used to extract the temperature dependence of the effective ultrasonic Gr\"uneisen parameter (UGP). For the first three crystals the temperature dependence of the UGP is qualitatively similar to that of the thermal GP squared, while its magnitude is in good agreement with calculations for the anisotropic continuum model. For neutron-irradiated quartz and for Ge-Si crystalline alloys the UGP should also be dependent on the concentration of defects or impurities. This dependence removes the discrepancies between the behavior of the thermal conductivity and of the ultrasonic attenuation of imperfect dielectrics. Independent available experimental evidence is presented in support of the present explanation of the above discrepancies.